Queueing theory

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Queue networks are systems in which single queues are connected by a routing network. In this image servers are represented by circles, queues by a series of retangles and the routing network by arrows. In the study of queue networks one typically tries to obtain the equilibrium distribution of the network, although in many applications the study of the transient state is fundamental.

Queueing theory is the mathematical study of waiting lines, or queues.[1] In queueing theory a model is constructed so that queue lengths and waiting time can be predicted.[1] Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

Single queueing nodes are usually described using Kendall's notation in the form A/S/C where A describes the time between arrivals to the queue, S the size of jobs and C the number of servers at the node.[5][6] Many theorems in queue theory can be proved by reducing queues to mathematical systems known as Markov chains, first described by Andrey Markov in his 1906 paper.[7]

Customers with high priority are served first.[19] Priority queues can be of two types, non-preemptive (where a job in service cannot be interrupted) and preemptive (where a job in service can be interrupted by a higher priority job). No work is lost in either model.[20]

Networks of queues are systems in which a number of queues are connected by customer routing. When a customer is serviced at one node it can join another node and queue for service, or leave the network. For a network of m the state of the system can be described by an m–dimensional vector (x1,x2,...,xm) where xi represents the number of customers at each node. The first significant results in this area were Jackson networks,[23][24] for which an efficient product-form stationary distribution exists and the mean value analysis[25] which allows average metrics such as throughput and sojourn times to be computed.[26]

If the total number of customers in the network remains constant the network is called a closed network and has also been shown to have a product–form stationary distribution in the Gordon–Newell theorem.[27] This result was extended to the BCMP network[28] where a network with very general service time, regimes and customer routing is shown to also exhibit a product-form stationary distribution. The normalizing constant can be calculated with the Buzen's algorithm, proposed in 1973.[29]

Networks of customers have also been investigated, Kelly networks where customers of different classes experience different priority levels at different service nodes.[30]

Another type of network are G-networks first proposed by Erol Gelenbe in 1993:[31] these networks do not assume exponential time distributions like the classic Jackson Network.

A system of inter-arrival time and service time showed exponential distribution, we denoted M.

λ：the average arrival rate

µ：the average service rate of a single service

P : the probability of n customers in system

n :the number of people in system

Let E represent the number of times of entering state n, and L represent the number of times of leaving state n. We have . When the system arrives at steady state, which means t, we have, therefore arrival rate=removed rate.

In discrete time networks where there is a constraint on which service nodes can be active at any time, the max-weight scheduling algorithm chooses a service policy to give optimal throughput in the case that each job visits only a single service node. In the more general case where jobs can visit more than one node, backpressure routing gives optimal throughput.

A network scheduler must choose a queuing algorithm, which affects the characteristics of the larger network.

Mean field models consider the limiting behaviour of the empirical measure (proportion of queues in different states) as the number of queues (m above) goes to infinity. The impact of other queues on any given queue in the network is approximated by a differential equation. The deterministic model converges to the same stationary distribution as the original model.[32]

Fluid models are continuous deterministic analogs of queueing networks obtained by taking the limit when the process is scaled in time and space, allowing heterogeneous objects. This scaled trajectory converges to a deterministic equation which allows us stability of the system to be proven. It is known that a queueing network can be stable, but have an unstable fluid limit.[33]